Theresa May smiles as she leaves parliament on Thursday.
Photograph: Tolga Akmen/AFP/Getty Images

Theresa May has agreed to set a timetable for her departure as prime minister in the first week of June, leading MPs to believe she will trigger a leadership contest before the summer.

Sir Graham Brady, the chair of the 1922 Committee of backbenchers, said she would agree a timetable for the election of a new leader after her Brexit legislation returned to parliament for a final attempt in the week of 3 June.

Another member of the 1922 Committee told the Guardian that May understood she would have to name a quick date for her departure if the withdrawal bill is voted down, with a leadership contest before the summer.

The MP said some Brexit supporters on the committee were disappointed that the prime minister was not forced to announce her departure immediately but this represented a “fair compromise”.

Brady’s announcement will intensify the leadership contest that has already been playing out among cabinet ministers and ambitious backbenchers for weeks. Boris Johnson, the former foreign secretary, was the latest to throw his hat into the ring on Thursday, saying he would “of course” go for it when there is a vacancy but there is a wide field of up to 20 candidates.

May met the 1922 committee’s executive on Thursday, which represents Tory backbenchers, after mutinous MPs demanded a specific timetable for her departure from No 10.

After a 90-minute meeting, Brady said: “We have agreed to meet to decide the timetable for the election of a new leader of the Conservative Party as soon as the second reading has occurred and that will take place regardless of what the vote is on the second reading – whether it passes or whether it fails.

“It was a very frank discussion, I tried to make sure that all the views represented on the executive were expressed and we had a very frank exchange with the Prime Minister.”

May still holds out a sliver of hope that her withdrawal bill legislation might not be voted down in its first week, enabling her to make the argument that she should not depart until she has seen the process of getting an agreement through.

However, the heavy hint that she will resign if the legislation is rejected once again is likely to incentivise even more Eurosceptic Tories to vote against it.

Labour has said it will not support it either without a formal deal involving a customs union and assurances that it cannot be unpicked by a future Conservative leader, meaning it is highly likely to be rejected. The cross-party Brexit talks are still ongoing but both sides will have to decide within days whether to continue when they are still some way apart on fundamental issues.

Regardless of whether there is a deal, there is speculation among some Labour and Tory backbenchers that No 10 could try to introduce the bill in a different form in an effort to get enough support for it to pass second reading at least. The idea of a “mini” or “bitesize” withdrawal agreement bill that could be later amended with different proposals is one possibility circulating.

MPs said they believed May had effectively acknowledged she will have to go if a vote on her Brexit deal fails for a fourth time, although some were suspicious that she might try to dodge the deadline again.

In that circumstance, 1922 committee members would be extremely likely to hold another vote on whether to have a no confidence motion in the prime minister, which would mean lifting party rules that prevent another one until December.

“If she cannot get the withdrawal bill through… supposing she still doesn’t resign then, that is the point there is likely to be a rule change or even the rules set aside. It would be unsustainable. And she understands that,” one 1922 committee member said.

Committee members differ on how long a leadership contest should take – with some warning it could take up to eight weeks – but all have agreed the parliamentary stages must be concluded by the summer recess.

With so many candidates expected to put their hats in the ring, and time needed to collect the nominations, it could take two to three weeks for MPs to whittle down the candidates to the final two that will be put to party members.

“The summer is our backstop, we want it done earlier than that, but it has to be done while parliament is sitting,” one committee member said. “You can only start MPs voting when you know who is putting themselves forward, you have to have hustings and the rules say votes should take place on Tuesdays and Thursdays. And there could be 20 people in the race.”

Only the last-placing candidate would be obliged to drop out of the race and organisers say they believe a crowded field could see contenders stay in the race for longer because numbers would be close between a number of them.

The prime minister had only committed previously to resign after passing the first stage of the Brexit process, before the negotiations on the future relationship begin.

But Downing Street had already previously hinted the prime minister sees the vote on the withdrawal agreement bill – scheduled for the week beginning 3 June – as make or break for her premiership and the deal she has negotiated. The week will be politically significant as Donald Trump’s state visit is also scheduled to take place and a by-election in the swing seat of Peterborough is due to happen on 6 June.

Asked when the prime minister should depart, Geoffrey Clifton-Brown, the 1922 treasurer, told Sky News: “Personally, the sooner the better, and that’s not being unkind to the prime minister. I just think the longer this goes on, it’s not in the nation’s interests, it’s not in the party’s interests. We’ve got European elections looming. Goodness knows what the results of that will be.

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“I think the genesis of this all started at the beginning of the negotiations,” he said. “If she had been much tougher on the negotiations – instead of allowing the Europeans to set the timetable – if she had said: ‘No, no, no, this is how we are going to do the negotiations, if you don’t like it, we’ll leave without a deal,’ then I think we would be in a much better position now.”

Should she still be in office, May will also face a no-confidence vote from party officials and members on 15 June at an extraordinary general meeting – though it is non-binding.